There is speculation that North Korea may also want payment for the return of the remains, our correspondent adds.
The rescheduled date, . this Sunday, July 15th.
Doubts over the prospect of a breakthrough were further exacerbated following the secretary of state Mike Pompeo's overnight visit to Pyongyang last week, during which a promised meeting with Kim failed to materialize.
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The U.S. Department of Defense estimates that North Korea is holding about 200 sets of remains from the some 5,300 American military personnel believed missing in the country and their recovery has always been among the most emotionally charged issues between the two sides.
It wasn't immediately clear what prompted the postponement.
At high-level talks last week, North Korea and the US confirmed they are still poles apart on the next steps to be taken in the process.
But Pompeo gave a rosier account saying the two sides had substantive discussions on next steps toward denuclearisation.
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In the letter Kim wrote to Trump he does not mention the word "nuclear" at all, but offers plenty of praise for the "unique" approach of Kim and Trump in improving relations between the two countries.
But there's growing skepticism in Washington over the Trump administration's engagement with North Korea.
Though Mr. Trump claimed on June 13 that North Korea was no longer a nuclear threat, just a day after his meeting with Mr. Kim, experts are not sure if North Korea is following through on the deal.
Their meeting - the first between a sitting USA president and North Korea, led to claims that the latter would work towards dumping its nuclear weapons.
Seoul's Defense Ministry said the remains of an unidentified allied soldier, presumably American, found in South Korea in 2016, will be sent to the United States.
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Bringing a formal end to the war, which ended in an armistice and not a peace treaty, is seen as the first step South Korea and the USA could take to guarantee the North Korean regime's security as they continue to push forward in negotiations for the North's denuclearisation.